无源控制理论分析

PassivecontroltheoryICarlesBatlleIIEURON/GEOPLEXSummerSchoolonModelingandControlofComplexDynamicalSystemsBertinoro,Italy,July18-222005ContentsofthislectureChangeofparadigmincontrol:fromsignalbasedtoenergybasedFitswellwiththePHDSmodelingapproachEnergybalancecontrolControlasinterconnectionCasimirfunctionsandthedissipationobstacleReferences„Ortega,R.,A.vanderSchaft,I.Mareels,andB.Maschke,Puttingenergybackincontrol.IEEEControlSystemsMagazine21,pp.18-33,2001.„Ortega,R.,A.vanderSchaft,andB.Maschke,Interconnectionanddampingassignmentpassivity-basedcontrolofport-controlledHamiltoniansystems.Automatica38,pp.585-596,2002.„Rodríguez,H.,andR.Ortega,Stabilizationofelectromechanicalsystemsviainterconnectionanddampingassignment.Int.JournalofRobustandNonlinearControl13,pp.1095-1111,2003.EnergybasedcontrolControlproblemshavebeenapproachedtraditionallyadoptingasignal-processingviewpoint.Veryusefulforlineartime-invariantsystems,wheresignalscanbediscriminatedviafiltering.Fornonlinearsystems,frequencymixingmakesthingsharder:veryinvolvedcomputationsverycomplexandenergycomsumingcontrolsareneededtosuppressthelargesetofundesirablesignalsMostoftheproblemstemsfromnotusinganyinformationaboutthephysicalstructureofthesystem.Wehavelearntfromthepreviouslecturesthatenergyplaysanessentialroleinthedescriptionofphysicalsystems.energybasedcontrolthecontrollershouldshapetheenergyofthesystem,andevenchangehowenergyflowsinsidethesystem.PassivityandenergybalancecontrolConsiderasystemwithstatesx∈Rn,inputsu∈Rmandoutputsy∈RmThemapu7→yispassiveifthereexistsastatefunctionH(x),boundedfrombelow,andanonnegativefunctiond(t)≥0suchthatZt0uT(s)y(s)ds|{z}energysuppliedtothesystem=H(x(t))−H(x(0))|{z}storedenergy+d(t)|{z}dissipatedmkλF(t)qpu=F,y=v=p/m˙q=pm˙p=F(t)−kq−λpm=Zt0µ˙p(s)+kq(s)+λmp(s)¶p(s)mds=Zt0F(s)v(s)dsZt0u(s)y(s)ds=µ12mp2(s)+12kq2(s)¶¯¯¯¯s=ts=0+λm2Zt0p2(s)ds=H(x(t))−H(x(0))+λm2Zt0p2(s)dsd(s)≥0H(x)=12mp2+12kq2ConsideranexplicitPHDSJT=−JRT=R≥0˙x=(J(x)−R(x))∂xH(x)+g(x)uy=gT(x)∂xH(x)=Zt0uTgT∂xHdτ=Zt0(gu)T∂xHdτZt0uTydτ=Zt0¡˙xT+(∂xH)TJ+(∂xH)TR¢∂xHdτ=H(x(t))−H(x(0))+Zt0(∂xH)TR∂xHdτ=Zt0∂τHdτ+Zt0(∂xH)TR∂xHdτdoesnotdependonthePHDSbeingexplicit!≥0PHDSarepassiveforu7→yandstoragefunctionHIfx∗isaglobalminimumofH(x)andd(t)0,andwesetu=0,H(x(t))willdecreaseintimeandthesystemwillreachx∗asymptotically.Therateofconvergencecanbeincreasedifweactuallyextractenergyfromthesystemwithu=−KdiywithKTdi=Kdi0Ifd≥0only,theninvariantsetshavetobeexaminedandLaSalletheoremhastobeinvoked.However,theminimumoftheenergyofthesystemisnotaveryinterestingpointfromanengineeringperspective.PhysicsAsengineers,wearenotreallyconcernedinknowingwhatNaturedoes,butratherinforcingNaturetodowhatwewant.Keyideaofpassivitybasedcontrol(PBC)usefeedbacku(t)=β(x(t))+v(t)sothattheclosed-loopsystemisagainapassivesystem,withenergyfunctionHd,withrespecttov7→y,andsuchthatHdhastheglobalminimumatthedesiredpoint.−Zt0βT(x(s))y(s)ds=Ha(x(t))IfWhyshouldthisbeastatefunction?thentheclosedloopsystemispassive(withinputv)andhasenergyfunctionHd(x)=H(x)+Ha(x)provethatIftheprecedingassumptionshold,thenThisiscalledtheEnergyBalanceEquation(EBE)Hd(x(t))|{z}closed-loopenergy=H(x(t))|{z}storedenergy−Zt0βT(x(s))y(s)ds|{z}suppliedenergyFor˙x=f+gu,y=hsystems,theEBEisequivalenttothePDE−βT(x)h(x)=µ∂Ha∂x(x)¶T(f(x)+g(x)β(x))Asanexample,considertheRLCcircuitx=µqφ¶stateu=Vcontroly=i=φLoutput∼V+LCR˙x=µx2/L−x1/C−x2R/L¶+µ01¶VH(x)=12Cx21+12Lx22ThemapV7→iispassivewithenergyfunctionanddissipationd(t)=Rt0RL2φ2(s)dsForV=V∗,theforcedequilibriaare(x∗1,0),withx∗1=CV∗IfwesetV=0,thenaturalequilibriais(0,0)ThePDEforenergybalancecontroltobesolvedisx2L∂Ha∂x1−µx1C+RLx2−β(x)¶∂Ha∂x2=−x2Lβ(x)Thenaturalenergyalreadyhasaminimumatthedesiredforcedequilibriax∗2=0weonlyneedtoshapetheenergywithrespecttox1wecantakeHaasafunctionofx1onlyDowereallyhavetosolvethis?β(x1)=−∂Ha∂x1(x1)Noequationatall:itdefinesthecontrol!TheonlyremainingtaskistochooseHasothatHdhastheminimumat(x∗1,0).Ha(x1)=12Cax21−µ1C+1Ca¶x∗1x1CaisadesignparametertobetunedforperformanceItcanbeseenthattheresultingHdhasaminimumat(x∗1,0)iffCa−Cu=−∂Ha∂x1(x1)=−x1Ca+µ1C+1Ca¶x∗1Considernowthisslightlydifferentcircuit˙x1=−1RCx1+1Lx2,˙x2=−1Cx1+V(t)∼V+RLCx∗1=CV∗,x∗2=LRV∗Onlythedissipationstructurehaschanged,buttheadmissibleequilibriaareoftheformThepowerdeliveredbythesource,Vx2/L,isnonzeroatanyequilibriumpointexceptforthetrivialonethisistheinfamousdissipationobstacle,whichwillreappearlaterthesourcehastoprovideaninfiniteamountofenergytokeepanynontrivialequilibriumpointControlasinterconnectionWewouldliketohaveaphysicalinterpretationoftheprecedingideasPlantandcontrollerastwophysicalsystemsexchangingenergyoveranetworkplantΣcontrollerΣcnetworkucycuyTheinterconnectionispowercontinuousifuTc(t)yc(t)+uT(t)y(t)=0∀tAsanexample,considerthestandardfeedbackinterconnectionplantΣcontrollerΣcucycuy−uc=yu=−ycItisclearlypowercontinuousucyc+uy=yyc+(−yc)y=0plantΣcontrollerΣcucycuy−vcvAssumewehaveoneofsuchinterconnections,andaddextrainputsu→u+v,uc→uc+vcLetΣandΣchavestatevariablesxandξ,andletthemapsu7→yanduc7→ycbepassive,withenergyfunctionsH(x)andHc(ξ